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Discrete Gradient Flow Approximations of High Dimensional Evolution Partial Differential Equations via Deep Neural Networks | | Emmanuil H. Georgoulis
; Michail Loulakis
; Asterios Tsiourvas
; | | Date: |
1 Jun 2022 | | Abstract: | We consider the approximation of initial/boundary value problems involving,
possibly high-dimensional, dissipative evolution partial differential equations
(PDEs) using a deep neural network framework. More specifically, we first
propose discrete gradient flow approximations based on non-standard Dirichlet
energies for problems involving essential boundary conditions posed on bounded
spatial domains. The imposition of the boundary conditions is realized weakly
via non-standard functionals; the latter classically arise in the construction
of Galerkin-type numerical methods and are often referred to as "Nitsche-type"
methods. Moreover, inspired by the seminal work of Jordan, Kinderleher, and
Otto (JKO) cite{jko}, we consider the second class of discrete gradient flows
for special classes of dissipative evolution PDE problems with non-essential
boundary conditions. These JKO-type gradient flows are solved via deep neural
network approximations. A key, distinct aspect of the proposed methods is that
the discretization is constructed via a sequence of residual-type deep neural
networks (DNN) corresponding to implicit time-stepping. As a result, a DNN
represents the PDE problem solution at each time node. This approach offers
several advantages in the training of each DNN. We present a series of
numerical experiments which showcase the good performance of Dirichlet-type
energy approximations for lower space dimensions and the excellent performance
of the JKO-type energies for higher spatial dimensions. | | Source: | arXiv, 2206.00290 | | Services: | Forum | Review | PDF | Favorites | |
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